Trigonometric identities provide a foundation that enables quick solutions to multiple mathematical expressions. This exercise demonstrates applying basic trigonometric ratios of sine, cosine and tangent when performing standard identity simplifications and verifications. The linkage among different trigonometric functions becomes possible through these identities, which enable us to show complex expressions in separate steps. Learning trigonometric connections helps students develop their logical reasoning skills, leading to advanced trigonometric abilities needed for engineering, architecture and physical science fields.
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Exercise 8.3 in the NCERT Solutions for Class 10 presents three important identities which state \(\sin^2 \theta + \cos^2 \theta = 1\) alongside \(1 + \tan^2 \theta = \sec^2 \theta\) and \(1 + \cot^2 \theta = \csc^2 \theta\). The relationships teach students methods to simplify problems through systematic verification of given expressions. The exercise functions as an essential tool for solidifying the knowledge described in NCERT Books. This exercise represents a fundamental requirement for solving trigonometric problems, which progress into height and distance calculations and advanced mathematical concepts.
Q 1: Express the trigonometric ratios
Answer:
We know that
(i)
(ii) We know the identity of
(iii)
Q 3: Choose the correct option. Justify your choice
(A) 1 (B) 9 (C) 8 (D) 0
Answer:
The correct option is (B) = 9
And it is known that
Therefore, equation (i) becomes,
Q 3: Choose the correct option. Justify your choice
(A) 0 (B) 1 (C) 2 (D) –1
Answer:
The correct option is (C)
we can write his above equation as;
Q 3: Choose the correct option. Justify your choice
Answer:
The correct option is (D)
The above equation can be written as;
We know that
therefore,
Answer:
We need to prove-
Now, taking LHS,
LHS = RHS
Hence proved.
Answer:
We need to prove-
Taking LHS;
= RHS
Hence proved.
[ Hint: Write the expression in terms of
Answer:
We need to prove-
Taking LHS;
By using the identity a 3 - b 3 =(a - b) (a 2 + b 2 +ab)
= RHS
Hence proved.
[ Hint : Simplify LHS and RHS separately]
Answer:
We need to prove-
Taking LHS;
Taking RHS;
We know that identity
LHS = RHS
Hence proved.
Answer:
We need to prove -
Dividing the numerator and denominator by
Hence Proved.
Answer:
We need to prove -
Taking LHS;
By rationalising the denominator, we get;
Hence proved.
Q 4: Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
Answer:
We need to prove -
Taking LHS;
[we know the identity
Hence proved.
Answer:
Given the equation,
Taking LHS;
[since
Hence proved
[ Hint : Simplify LHS and RHS separately]
Answer:
We need to prove-
Taking LHS;
Taking RHS;
LHS = RHS
Hence proved.
Answer:
We need to prove,
Taking LHS;
Taking RHS;
LHS = RHS
Hence proved.
Also read-
1. Understanding basic trigonometric identities: You must learn the essential three trigonometric identities and understand their value for problem simplification and solution.
2. Verifying trigonometric identities: Perform proofs which establish two sides of an equation to be equal by using proper transformations in combination with trigonometric relations.
3. Developing logical and proof-solving skills: Step-by-step logical verification belongs to the set of skills that students need to develop for checking identities and producing mathematical proofs.
4. Preparing for advanced applications: Construct a strong foundation to solve problems utilising height-distance relationships alongside trigonometric equation methods, as well as actual angle measurement scenarios.
Check Out-
Students must check the NCERT solutions for class 10 of the Mathematics and Science Subjects.
Students must check the NCERT Exemplar solutions for class 10 of the Mathematics and Science Subjects.
Frequently Asked Questions (FAQs)
No, calculators are not allowed in Class 10 board exams, so you must rely on your memorised values and identities.
Yes, Exercise 8.3 is very important as identity-based questions frequently appear in CBSE Class 10 board exams
You should at least remember the fundamental identities and standard angle values to solve Exercise 8.3 effectively.
On Question asked by student community
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